Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Applications of Calculus.
Given an equation - Test Yourself 2 - Solutions.


 

Minimum value. 1. (i)

(ii)Point of inflection.

x 0 1 2
y" -6 0 6

So a change of concavity. POI is at (1, -2).

(iii)

(iv) At x = -2, y = -20 which is less than the y value at x = 2.

So the minimum value in the domain [-2, 3]
is -20.

NOTE: Do NOT give the x value - it is just the minimum value of the curve!!
  2.  
Maximum value 3.  
  4.  
Curve increasing/decreasing. 5. (i) For the curve y = 3x - x3, find the stationary points and determine their nature.

(ii) For what values of x is the curve increasing?

Answer. (i) Maximum at (1,2);
minimum at (-1, -2).
(ii) Increasing for -1 < x < 1.

  
  6.  
Concavity 7. A function is defined by f(x) = x3 - 3x2 - 9x - 22.

(i) Find the coordinates of the turning points of the function f(x) and determine their nature.

(ii) Find the coordinates of the points of inflection.

(iii) Sketch the graph of y = f(x) showing the main features.

(iv) For what values of x is the graph of y = f(x) concave up?

Answer. (i) TPs at (-1, 27) max and
(3, -5) min;
(ii) POI at (1, 11).
(iv) x > 1.		  
  8.  
Points of inflexion. 9.  
  10.  
Number of solutions. 11.  
  12.